Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just positive integers. The function is 1 if the variables are equal, and 0 otherwise:

where the Kronecker delta δij is a piecewise function of variables i and j. For example, δ1 2 = 0, whereas δ3 3 = 1.

The Kronecker delta appears naturally in many areas of mathematics, physics and engineering, as a means of compactly expressing its definition above.

In linear algebra, the n × n identity matrix I has entries equal to the Kronecker delta:

where i and j take the values 1, 2, ..., n, and the inner product of vectors can be written as

The restriction to positive integers is common, but there is no reason it cannot have negative integers as well as positive, or any discrete rational numbers. If i and j above take rational values, then for example

This latter case is for convenience.

Properties

The following equations are satisfied:

Therefore, the matrix δ can be considered as an identity matrix.

Another useful representation is the following form:

This can be derived using the formula for the finite geometric series.

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